Evolutionary implications of SARS-CoV-2 vaccination for the future design of vaccination strategies

Similarly to HIV, HCV, and influenza virus, SARS-CoV-2 is perpetually acquiring new mutations in its genome, with an average substitution rate of (0.7-1.1) × 10⁻³/year/site⁴⁶ (Fig. 1). SARS-CoV-2 evolution is especially fast in the spike protein^47,48,49,50. Three major reasons account for the rapid evolution in viral receptor proteins, such as the spike of SARS-CoV-2, hemagglutinin of influenza, and gp120 of HIV. Firstly, the spike has receptor-binding motifs that affect the transmission, and their evolution leads to an increase in virus fitness. This may play a role in the emergence of VOCs with enhanced transmissibility^9,10,11,51. Secondly, it contains epitopes, regions that are very important for the immune response because of their involvement in the binding of antibodies that can neutralize the virus. Mutations in epitopes, in addition to the waning of antibodies, are a major factor that limits viral recognition by the immune system and, hence, the durability of protection against infection^47,52,53. Thirdly, these epitope regions have evolved to have low physiological constraints on mutation (low mutation cost) because they serve primarily as highly-variable decoys for antibodies. Had they another important function for a virus, they would be conserved. For example, the receptor-binding site has a function and is conserved, because it hides between the protruding variable regions to prevent antibody binding. This is the case for both the influenza virus receptor (hemagglutinin) and HIV receptor (gp120)^54,55,56,57.

Blue rectangles show the intervals of the median values for the most rapidly and most slowly evolving subtypes of HIV, HCV, influenza virus, and SARS-CoV-2 for the full genome^{46,161,162,163}. Red rectangles show the rates for the proteins targeted by neutralizing antibodies^{47,54,164,165}. HIV data are multiplied by 3.

In the next sections, we focus on the population-level evolution in neutralizing antibody epitopes related to immunity, vaccination, and viral recognition and on escape mutations that occur during transmission chains of acute infections. We postpone until the final section the discussion of the evolution outside of epitopes and of rare chronic infections, where escape mutations can accumulate within one individual.

In what follows, we assume—and this is the only essential assumption to be tested in the future experiments on which our discussion relies—that for SARS-CoV-2 the cost of mutations in antibody-neutralizing regions to virus replication ability is as small as that for influenza virus and HIV. We make this assumption because the structure of antibody-binding sites on the spike protein of SARS-CoV-2 is similar to the structures on gp120 of HIV and hemagglutinin of influenza. It comprises several protrusions of similar lengths covered in sugars, located far from the receptor-binding site, and serving as targets for antibodies. Then, the selection pressure for these viruses to escape is reasonably expected to be of the same order of magnitude. In the general case, the final cost of mutation limits the antigenic escape^58,59,60, so this assumption remains to be tested in the future ly. Most research has, so far, focused on mutations causing the emergence of VOC, and we hope that this discussion will bring the focus to finding new epitope variants for SARS-CoV-2 as it happened for influenza and HIV.

The maximal vaccination coverage supported by public health authorities for controlling the pandemic had important consequences. In the shorter term of a few months, this approach facilitated a decrease in the number of infections, helped to unload hospitals, and to reduce COVID-19-related mortality. In the longer term, as we show in this Perspective, mass vaccination potentially further accelerates the rapid evolution of epitope regions. Let us start, however, with unvaccinated populations.

Virus evolution in epitopes in the absence of vaccination: the Red Queen effect

In the absence of vaccination, virus evolution in epitopes at the population level is driven by the immune response in individuals recovered from natural infections whose number gradually accumulates in the population. The process is observed for other respiratory viruses, including influenza, and it has been investigated in detail in genomic, immunological, and bioinformatic studies^{61,62,63,64,65,66,67}. Its evolutionary dynamics has been interpreted and predicted using mathematical models^{54,68,69,70,71,72,73,74,75}. The perpetual immune escape in epitopes is the reason why seasonal influenza is not becoming extinct but persists among humans. In order to avoid extinction, the influenza virus has to continue mutating to distance itself genetically from the immune response accumulating in the population. Such adaptation of an organism facing an evolving opposing species is termed “The Red Queen effect” (Fig. 2).

The effect bears the name of the Red Queen’s race from the novel “Through the Looking-Glass” by Lewis Carroll. As the Red Queen told Alice: “Now, here, you see, it takes all the running you can do, to keep in the same place.” Similarly, the evolution of SARS-CoV-2 in epitopes at the population level is driven by the immune response in individuals recovered from natural infection or vaccinated, whose number gradually accumulates in the population. In order to avoid extinction, the virus has to mutate perpetually to distance itself genetically from this immune response accumulating in the population. Image source: The picture is a modified version of the illustration by Sir John Tenniel from Lewis Carroll’s “Through the Looking-Glass”, 1871, downloaded from https://www.alice-in-wonderland.net/resources/pictures/through-the-looking-glass/.

The persistent epitope evolution is observed in seasonal human coronaviruses and SARS-CoV-2^47,48,49,50. However, the substitution rate in the spike protein of SARS-CoV-2, which is three times faster than that of influenza’s hemagglutinin, is unprecedented for an acute respiratory virus⁴⁷. The consequences of this process become clear if we introduce two quantities describing the potential of a virus to spread in a population, the basic and effective reproduction numbers, R₀ and R_e (Box 2). Both measure the Darwinian viral fitness on the population level defined as the number of individuals infected by one individual. They incorporate several host-level factors, including the number of virions per cell, infectivity, transmission dose, the immune response in a host, and transmission bottlenecks. The former, \({R}_{0}\), denotes the average number of newly infected individuals caused by one infected individual in a fully susceptible (naïve) population. The latter, \({R}_{e}\), denotes the average number of new infected individuals caused by one infected individual in a population where some individuals are immune. Both \({R}_{0}\) and \({R}_{e}\), are defined for given public measures in place, such as lockdowns and various restrictions. They differ, by definition, only due to the immune memory accumulated in a population. Therefore, public health measures reduce \({R}_{0}\) and \({R}_{e}\) by the same factor.

\({R}_{0}\) is larger than 1 for the original SARS-CoV-2 variant (range 1.9 – 6.5)⁷⁶ and for Alpha, Delta, Gamma, Omicron and other VOCs^8,9,11,77. This condition means that each infected individual transmits the virus to more than one other individual which initially leads to an exponentially increasing number of infected individuals. With more and more individuals being infected and becoming immune, \({R}_{e}\) would keep decreasing until, with no other changes, transmission would become negligible.

However, as a result of the ongoing evolutionary escape from the immune response after natural infection, as well as due to mutations outside of the epitopes including the receptor-binding domain, this outcome is altered to the stationary process with seasonal oscillations in the virus prevalence due to seasonality in transmission. At the beginning of each seasonal epidemic, \({R}_{e}\) is larger than \(1.\) As the virus infects more people who acquire immunity, \({R}_{e}\) becomes lower than 1. When averaged over seasonal epidemics of several years, \({R}_{e}\) is close to 1. Thus, each infected individual transmits the virus, on average, to one individual keeping the number of the infected individuals averaged over a long period of time approximately constant^73,74,75. This stationary process with seasonal oscillations is established after most people have already been infected at least once.

Models of virus evolution in epitopes at the population level

Mathematical models connect the initial assumptions to predictions in the most accurate and reproducible way. A key benefit of using models is that they allow uncertainty to be quantified and to conduct scenario analyses based on a range of initial assumptions. An important observable quantity predicted by mathematical models and measured from viral genome sequence data is the speed of antigenic evolution or substitution rate, \(V\), defined as the rate of accumulation of non-synonymous mutations in neutralizing antibody epitopes (Box 2). Using mathematical models, \(V\) can be expressed in terms of several parameters, namely the number of infected individuals, \({N}_{{{\inf }}}\), and mutation rate, \({U}_{b}\), defined as the probability of an escape mutation per transmission in epitopes (Box 2). The latter is a composite parameter that depends on the conditions in an individual host including the virus replication error rate in a cell and the host immune response creating natural selection for immune escape mutants^78,79. Substitution rate, \(V\), also depends on the virus recombination rate and the transmission advantage of an escape mutant, \(s\). The form of the relationship between \(V\) and parameters \({N}_{{{\inf }}}\), \({U}_{b}\), \(s\) may vary depending on the most important factors of evolution and the values of these parameters.

In the simplest case, when the product \({N}_{{{\inf }}}{U}_{b}\) is much less than 1, immune escape mutations emerge and spread through the population one at a time (Fig. 3a). In this case, the substitution rate, \(V\), is proportional to \({N}_{{{\inf }}}{U}_{b}.\) This assumption is implicitly built into some epidemiological models of seasonal influenza and SARS-CoV-2^78,80,81,82.

Immune population consists of individuals who recovered from natural infection and who were vaccinated. Using mathematical models, population-level viral substitution rate in epitopes, \(V\), can be expressed in terms of the effective number of infected individuals, \({N}_{{{\inf }}}\), and mutation rate, \({U}_{b}\), defined as the probability of an escape mutation per transmission in epitopes. a At a low mutation rate per population per epitope per transmission, \({{N}_{{{\inf }}}U}_{b}\ll 1\), the supply of immune escape mutations is low. As a consequence, escape mutations spread in the population one at a time. The population-level substitution rate \(V\) (black line) is proportional to the immune selection pressure in a population \({s}_{{{{{{\rm{tot}}}}}}}\) (gray line) and the total number of infected individuals where escape mutants can emerge \({N}_{{{\inf }}}\) (red line). The dependence of \(V\) on the proportion of the immune population has a maximum (black line). b At a high mutation rate, \({{N}_{{{\inf }}}U}_{b}\gg 1\), many escape mutations at different positions of the viral genome spread in the population at almost the same time and compete with each other for human hosts. The substitution rate \(V\) (black line) is proportional to selection pressure \({s}_{{{{{{\rm{tot}}}}}}}\,\)(gray line) and weakly depends on \({N}_{{{\inf }}}\) (red line), so that \(V\) increases monotonically with the proportion of the immune population. a, b Blue people are recovered; orange, yellow, and red people are infected with viral variants with different immune escape mutations (genomes below). The evolution of seasonal influenza and SARS-CoV-2 is compatible with b and not with a (see text).

However, if the product \({N}_{{{\inf }}}{U}_{b}\) is much larger than 1, mathematical models predict that mutations occurring at different positions of the viral genome are concurrent in time, and the approximation of independent sweeps does not apply anymore due to strong interference between different mutations^{83,84,85,86,87,88,89,90,91,92}. This linkage interference creates several effects such as interference between emerging clones (Fisher-Müller effect)^93,94, which is equivalent⁹⁵ to Hill–Robertson effect⁹⁶, i.e., the decrease of selection effect at one locus due to selection at another locus, as well as various genetic background effects. All of these effects are directly observed for seasonal influenza and many other viruses^54,93,97,98 (Fig. 3b). The linkage interference effects slow down virus evolution by orders of magnitude and change the way the substitution rate^{83,84,85,86,87,88,89,90,91,92}, \(V\), and the statistics of phylogenetic trees^99,100 depend on the number of infected individuals, \({N}_{{{\inf }}}\), mutation rate, \({U}_{b}\), and recombination rate. We refer to this situation as “multi-locus regime.”

SARS-CoV-2 genome has hundreds of evolving sites¹⁰¹. The virus demonstrates, on average, more than two substitutions per month, or 1.1 \(\times\) 10⁻³ substitutions per site per year⁴⁶, and modest intra-host diversity^102,103. Fast evolution is common for RNA viruses because they lack proofreading enzymes, their mutation rates are relatively large¹⁰⁴. They all fall in the range of 10⁻⁶ to 10⁻⁴ per nucleotide per replication. To estimate \({U}_{b}\), we have to multiply this mutation rate by the size of the infected population and the length of the antibody binding region. For influenza A, the mutation rate per transmission event per antibody binding region is estimated at 3 × 10^−4 64,73. For SARS-CoV-2, the population-level mutation rate is expected to fall within the same order of magnitude. Therefore, the multi-locus regime (\({N}_{{{\inf }}}\) \({U}_{b}\) > 1) applies if more than N_inf = 10,000 infected individuals are present in a population, which is the case during a pandemic wave in a large city. The fact that \({N}_{{{\inf }}}\) \({U}_{b}\) > 1 for influenza A H2N3, which falls into the multi-locus regime, was demonstrated using sequence data⁵⁴.

Thus, SARS-CoV-2 evolution can be described by multi-locus models, which have been studied intensely over the last two decades using the methods of statistical physics^{83,84,85,86,87,88,89,90,91,92,99,100}. Their exact predictions for the substitution rate, \(V\), vary depending on the specific evolutionary factors taken into consideration. Genetic variants arise by random mutation but are subsequently amplified or suppressed by natural selection, with random genetic drift and linkage as additional stochastic factors. However, all these models demonstrate that \(V\) depends weakly on both the number of currently infected individuals, \({N}_{{{\inf }}}\), and mutation rate, \({{U}_{b}}\). More specifically, \(V\) is linearly proportional to \({{{\log }}{N}}_{{{\inf }}}\) and increases logarithmically with \({{U}}_{{{{{{\rm{b}}}}}}}\) as well (Fig. 3b).

The multi-locus models predict that the average substitution rate, \(V\), is proportional to the selection pressure that describes the change of viral fitness due to mutation^{83,84,85,86,87,88,90,92}. The selection pressure in epitopes due to immunity accumulating in an unvaccinated host population has been investigated in several modeling studies^73,74,75. These models account for the immune response after natural infection, and their most important result is the general expression for the average transmission advantage of an escape mutation in an epitope created by natural immunity in recovered individuals. This quantity, analogous to the average selection coefficient in evolutionary theory, \(s\), is expressed in terms of only two viral parameters, one epidemiological and another immunological. The epidemiological parameter is \({R}_{0}\), which we introduced earlier. The immunological parameter is the cross-immunity distance, \(a\), defined as the value of the genetic distance between the infecting virus and the virus from which the individual recovered previously, at which the host susceptibility to the virus is half of its maximum value in the fully susceptible (naïve) individual. The cross-immunity distance, \(a\), is a composite parameter that combines the cross-reactivity of antibodies with their total number left after infection. The average transmission advantage of an escape mutation in an epitope has the general form^73,74,75

where the function \(f\) grows slower than a linear function, and \(f(1)=0\). Equation 1 means that the average transmission advantage of an escape mutation in an epitope, \(s\), is lower for a larger cross-immunity distance, \(a\), i.e., when antibodies are more broadly neutralizing (epitope binding by antibodies is less sensitive to mutations in epitope). Equation 1 also states that \(s\) grows with the basic reproduction number, \({R}_{0}\). Indeed, \({R}_{0}\) defines the viral transmission potential in the naïve population and, hence, sets the scale for transmission changes with mutation.

The specific form of function \(f\) in Eq. 1 depends on the properties of the natural immune memory and cross-recognition, and varies among analytic multi-locus models^73,74,75. While these studies differ in the description of natural immunity and, hence, in the accuracy of the predictions, they all agree on two important predictions. Firstly, these studies produce expressions for the selection pressure driving the antigenic wave in the form of Eq. 1. Secondly, corroborating earlier numerical findings^69,70, the evolution of the virus and immune memory in a population is predicted to be a traveling wave in the genetic space that has a quasi-one-dimensional shape.

Cited studies^73,74,75 assumed that the genetic distance is linearly proportional to antigenic distance. An additional complication is that different mutations in epitopes have a variable effect on epitope binding. Some mutations make a large change in the free energy of binding, and some mutations are almost neutral. This means that the genetic distance and antigenic distance are not linearly proportional. In terms of the phenotype landscape, it has a rugged component. This fact has been investigated in detail using two-dimensional antigenic maps for influenza^61,70 and is of major importance when short-term evolution is studied. In the long-term evolution, an effective ratio of antigenic to genetic change can be successfully described by studies with the averaged-out cross-immunity function^71,73,74,75.

Effect of vaccination on viral evolution in epitopes at the population level

Vaccination slows down transmission by decreasing the probability of infection of vaccinated individuals and by decreasing the viral load in such individuals should they become infected. Put simply, vaccination also makes individuals less infectious. Some studies argued that, because the virus is poorly transmitted in the vaccinated population, the reduction in the effective reproduction number by vaccination will also slow down the evolution of epitopes⁷⁸. This prediction would be correct under the assumption that \({R}_{e}\) was reduced below 1 very rapidly and remained at that level for a period of time long enough for the virus to be eradicated. The situation is analogous to the successful antiretroviral therapy in an HIV-infected individual. The therapy not only suppresses the number of infected cells by several orders of magnitude, but also strongly impedes within-host evolution, because it reduces the initial reproduction number below 1 very rapidly. An example at the population level is vaccination against childhood infections like measles-mumps-rubella which has a very high efficacy against infection and eradicates the virus in a population with sufficiently high vaccination coverage.

Unfortunately, due to the combination of factors such as vaccine efficacy against infection below 100%, incomplete vaccination coverage, mutations in the receptor-binding region, and pre-existing epitope mutants, \({R}_{e}\) of SARS-CoV-2 does not fall fast enough. It soon rebounds above 1 allowing for the virus evolution in a population to continue. In situations when vaccination does not have the rapid eradicating effect, it does not slow down but, on the contrary, applies additional selection pressure due to the additional immune memory cells it creates, similarly to the selection pressure from natural infection^70,73,74,75. If the cost of mutations is low enough, this immune selection pressure will further accelerate virus evolution in antibody-binding regions^58,59,60. The effect is analogous to the case of suboptimal therapy in an HIV-infected individual that selects drug-resistant mutants. These mutants exist in very small quantities before therapy and become dominant in a patient within weeks of failing therapy. Highly-active drug cocktails have solved this problem. A similar dichotomy for neutralizing and non-neutralizing vaccines was predicted for the evolution of virulence¹⁰⁵. Note that since we consider the dynamics at the population level, the effect discussed here will be the same both for a “leaky” vaccine, where all susceptible individuals have reduced susceptibility to infection after vaccination, and for an “all or nothing” vaccine, where a proportion of susceptible individuals are completely protected by vaccination.

In the case of the epitope evolution of SARS-CoV-2 in a population, vaccination adds the immunity of vaccinated individuals to the natural immunity of recovered individuals, thus potentially favoring immune escape mutations even more. Using an equestrian analogy, vaccination spurs the Red Queen into a full gallop. Thus, the cost of the short-term decrease in the number of infections is the spread of new epitope mutations in the future. The extent to which vaccination accelerates evolution in antibody-binding regions depends on vaccination frequency, inter-dose period, molecular design of vaccines, and details of immunological and evolutionary dynamics. To avoid confusion, we emphasize that we discuss here the evolution of epitopes only and not, for example, the emergence of VOC related to mutations in other regions. Before Omicron and its descendants, all VOCs had emerged before the mass rollout of vaccination as inferred from phylogenetic analyses^106,107. Vaccination was probably not involved as a selection pressure in their genesis.

Just to illustrate the potential magnitude of the Red Queen effect, let us make a ballpark estimate. In the case of high vaccination coverage combined with non-pharmaceutical interventions, we observe a sharp short-term drop in the total number of currently infected individuals, \({N}_{{{\inf }}}\). The long-term impact of the decrease in the number of infected individuals on viral evolution in epitopes can be found from the substitution rate, \(V\). As already mentioned, all multi-locus models demonstrate that \(V\) depends logarithmically on both \({N}_{{{\inf }}}\) and mutation rate, U_b¹⁰⁸.

For example, if vaccination has decreased the number of infected individuals in a city 100-fold from 10,000 to 100, the corresponding decrease in the virus substitution rate is only two-fold. This is a drastically different result from a 100-fold reduction expected from the simpler models assuming rare immune escape mutations where the substitution rate is linearly proportional to product \({N}_{{{\inf }}}\) \({U}_{b}\). The same consideration applies to parameter \({U}_{b}\) which has been argued to depend on the vaccination dose in a vaccinated host^78,79. The effect of a change in \({U}_{b}\) on \({V}\) is very small (i.e., logarithmic). At the same time, the substitution rate, \(V\), is linearly proportional to the selection pressure, \(s\), created by the individuals with natural or vaccine-induced immunity^73,74,75, Eq. 1 (Fig. 4).

The substitution rate in epitopes is shown in the presence (three higher lines) and in the absence (three lower lines) of recovered individuals as the proportion of the vaccinated population increases. Red, green, and blue lines correspond to vaccine efficacy in inducing protection against infection equal, lower, and higher than the natural immune response. We assume the absence of fully susceptible (naïve) individuals.

Thus, vaccination creates two opposing effects on the Red Queen adaptation in epitopes. One effect comes from the decrease in transmission rate due to partial immune protection and lower virus amount transmitted to another individual. This effect creates a positive selection pressure for resistant mutations. The opposite effect of vaccination comes from the decrease in the mutation rate within a host, due to lower viral load. The first effect wins, because the adaptation rate is linearly proportional to selection pressure, \(s\), and weakly depends on the mutation rate, \({U}_{b}\) (Fig. 3b).

Several studies^78,80,81 use standard compartmental infectious disease models and arguments following those to predict that vaccination can decrease vaccine escape by reducing the number of infectious individuals. We would like to point out that the aforementioned studies lack evolutionary dynamics because it is not built into these simplified models. For example, ref. ⁸⁰ assume the vaccine escape pressure to be proportional to the number of vaccinated individuals. At best, models of this type correspond to the case when immune escape mutations emerge and spread through the population one at a time (single-site approximation or independent-locus models; Fig. 3a). As we explain above, this is the case for unrealistically-small population sizes. These models do not include proper treatment of evolutionary dynamics and disregard linkage effects existing between mutations at multiple sites, because such models are technically more difficult to handle. Here, we use the modern theory of multi-locus virus evolution that takes into account clonal interference, genetic background effect, and other linkage effects, random genetic drift, and natural selection arising due to immune memory. From the evolutionary viewpoint, multi-locus models are closer to reality than independent-locus models. At the same time, based on strong genetic variation in the antibody epitopes in the spike protein of SARS-CoV-2, we assumed that mutations in these regions have a low cost, by analogy with mutations in hemagglutinin of influenza. The validity of these assumptions remains to be tested directly in the future.

Estimation of the vaccination effect on the substitution rate in epitopes

The modeling results obtained for natural immunity^73,74,75 can be generalized for immunity induced by vaccination. Due to the immune response in vaccinated individuals the total selection pressure of escape, denoted \({s}_{{{{{{\rm{tot}}}}}}}\), is increased by an additional term, denoted \({s}_{{{{{{\rm{vac}}}}}}}\), as follows

where \(s\) is given by Eq. 1, and \({s}_{{{{{{\rm{vac}}}}}}}\) corresponds to the effect of vaccination averaged over time. Because the substitution rate, \(V\), is linearly proportional to the selection pressure, Eq. 1, it increases linearly with the vaccination term, \({s}_{{{{{{\rm{vac}}}}}}}\). The selection pressure due to vaccination, \({s}_{{{{{{\rm{vac}}}}}}}\), depends on the type of vaccine and the epitopes it exposes to the immune system. It also depends on vaccination frequency and the period between vaccine doses which determine the average genetic distance between the vaccine vector and the currently circulating viral variant. The more recent the vaccine vector is the stronger selection pressure to escape immune response in vaccinated individuals it exerts on the virus. This relationship is determined by an important immunological parameter, termed “cross-immunity function” that represents the decrease in the host susceptibility with the genetic distance \(x\) between the infecting virus and a virus from which the individual recovered in the past^68,73,74,75. The form of the cross-immunity function determines the cross-recognition of epitope variants and may vary among epitopes, host organisms, and viruses. Reconstructing this function in each case remains a challenge but one can estimate from viral genome sequence data the cross-reactivity half-distance, i.e., the genetic distance where the host susceptibility is half of that in a naïve individual. For example, for influenza A H3N2 the cross-reactivity half-distance of 15 amino acid substitutions has been inferred for humans⁷³ and measured for equines¹⁰⁹. For SARS-CoV-2, these estimates are still to be determined.

To get an idea about the magnitude of the effect of vaccination on the substitution rate in epitopes, let us consider a simple example. We assume (i) a vector vaccine similar to Sputnik V or that generated by AstraZeneca that has the entire spike protein with the same epitopes as the natural virus; (ii) vaccine vector is based on a virus variant similar to a variant that infected a typical individual recovered from natural infection; (iii) immunity after natural infection and vaccination are similar. Suppose that 1% of the population are naïve susceptible, 9% of the population had a natural infection and recovered, and the remaining 90% of individuals were vaccinated. With these assumptions, we have 90/9 = 10-fold more individuals with vaccine-induced than natural immunity, and hence \({s}_{{{{{{\rm{vac}}}}}}} \sim 10{s}\). If \({{\log }}\,{N}_{{{\inf }}}\) decreases due to vaccination by the factor of 2, as we estimated above, the substitution rate, which is proportional to both \({s}_{{{{{{\rm{tot}}}}}}}\) and \({{\log }}\,{N}_{{{\inf }}}\), increases by the factor of ~ \(1/2\,\times \,10\,=\,5\).

If the ratio of the recovered and vaccinated was different from 1:10, the effect on the substitution rate in epitopes would differ from this estimate. The schematic of the dependence of the substitution rate in epitopes for varying proportions of vaccinated and recovered population is shown in Fig. 4. The case of some African countries which had multiple waves of SARS-CoV-2 infection and hardly any vaccination corresponds to a small proportion of the vaccinated population (extreme case: 0%). The case of some European countries with massive vaccination efforts corresponds to a very high proportion of the vaccinated population (extreme cases 100% achieved in, e.g., the Portuguese elderly). Seasonal influenza with its annual vaccination campaigns in many countries with temperate climates would also correspond to a rather small proportion of the vaccinated population, as compared to the global mass vaccination against SARS-CoV-2 realized within 2 years.

We also assumed that the effects of vaccinal and natural immune response on selection pressure are the same. In fact, such symmetry is unlikely, because the number of immune memory cells against an epitope induced by vaccination and natural infection may differ. Figure 4 shows schematically how the substitution rate in epitopes changes with the relative protection (and hence, the selection pressure on epitopes) rendered by the vaccine as compared to the natural infection (compare the red line with the green and blue lines). In addition, vaccines are composed of a section of the spike protein, and the immune system generates antibodies against other viral proteins as well. Thus, the effect of vaccination on the substitution rate in an epitope can be either stronger or weaker than the effect of the natural immune response. Furthermore, the genomic regions where evolution is accelerated will also differ between vaccinal and natural responses. A detailed study based on a mathematical model and immunological data is required to calculate the acceleration of evolution in various epitopes.

For the sake of simplicity, in our example, almost everyone is either recovered or vaccinated, and the overlap between the two groups is neglected. In reality, there are many people who first recovered from natural infection and then got vaccinated, and vice versa. The overlap might lead to a further increase in the speed of evolution, due to the combined effect of natural and induced immunity, however, the interaction between the two is not trivial.

To summarize, we arrived at very different conclusions regarding the effect of vaccination on the speed of antigenic evolution compared to the previous work^78,80,81 by exploiting the similarities in the molecular structure of antibody-binding regions of different viruses and using the modern theory of multi-locus evolution driven by the immune response. The above example illustrates that, despite the reduction of viral transmission by vaccination, viral evolution in neutralizing antibody epitopes may be accelerated by vaccination several-fold. The transient decrease in the number of infections is outweighed by a stronger immune pressure to change. The cost of the transient reduction in virus circulation is the emergence of more transmissible escape mutants and, hence, a higher number of infected individuals in the population in the future.